prediction of rainfall time series using modular

Journal of Hydrology, Vol. 389, No. 1-2, 2010, pp. 146-167

1

Prediction of Rainfall Time Series Using Modular Artificial Neural Networks Coupled 1

with Data Preprocessing Techniques 2

C. L. Wu1,2, K. W. Chau1,*, and C. Fan3 3

1 Dept. of Civil and Structural Engineering, Hong Kong Polytechnic University, 4

Hung Hom, Kowloon, Hong Kong, People’s Republic of China 5

6

2 Changjiang Institute of Survey, Planning, Design and Research, 7

Changjiang Water Resources Commission, 8

430010, Wuhan, HuBei, People’s Republic of China 9

10

3 Department of Civil Engineering, Ryerson University, 11

350 Victoria Street, Toronto, Ontario, Canada, M5B 2K3 12

13

*Email: [email protected] 14

15

ABSTRACT 16

This study is an attempt to seek a relatively optimal data-driven model for rainfall forecasting from three aspects: 17

model inputs, modeling methods, and data preprocessing techniques. Four rain data records from different 18

regions, namely two monthly and two daily series, are examined. A comparison of seven input techniques, 19

either linear or nonlinear, indicates that linear correlation analysis (LCA) is capable of identifying model inputs 20

reasonably. A proposed model, modular artificial neural network (MANN), is compared with three benchmark 21

models, viz. artificial neural network (ANN), K-nearest-neighbors (K-NN), and linear regression (LR). 22

Prediction is performed in the context of two modes including normal mode (viz., without data preprocessing) 23

and data preprocessing mode. Results from the normal mode indicate that MANN performs the best among all 24

four models, but the advantage of MANN over ANN is not significant in monthly rainfall series forecasting. 25

Under the data preprocessing mode, each of LR, K-NN and ANN is respectively coupled with three data 26

This is the Pre-Published Version.


2

preprocessing techniques including moving average (MA), principal component analysis (PCA), and singular 27

spectrum analysis (SSA). Results indicate that the improvement of model performance generated by SSA is 28

considerable whereas those of MA or PCA are slight. Moreover, when MANN is coupled with SSA, results 29

show that advantages of MANN over other models are quite noticeable, particularly for daily rainfall 30

forecasting. Therefore, the proposed optimal rainfall forecasting model can be derived from MANN coupled 31

with SSA. 32

KEYWORDS 33

Rainfall prediction, Modular artificial neural network, Moving Average, Principal component analysis, Singular 34

spectral analysis, Fuzzy C-Means clustering, K-nearest-neighbors 35

36

1. Introduction 37

Accurate and timely rainfall forecasting is crucial for reservoir operation and flooding 38

prevention because it can provide an extension of lead-time of the flow forecasting, larger 39

than the response time of the watershed, in particular for small and medium-sized 40

mountainous basins. 41

Many studies have been conducted for the quantitative precipitation forecasting using 42

diverse techniques including numerical weather prediction models and remote sensing 43

observations (Yates et al., 2000; Ganguly and Bras, 2003; Sheng, et al., 2006; Doomede et al., 44

2008), statistical models (Chu and He, 1995; Chan and Shi, 1999; DelSole and Shukla, 2002; 45

Munot, 2007; Li and Zeng, 2008; Nayagam et al., 2008), chaos theory-based approach 46

(Jayawardena and Lai, 1994), K-nearest-neighbor (K-NN) method (Toth et al, 2000), and soft 47

computing methods including artificial neural network (ANN), support vectors regression 48

(SVR) and fuzzy inference system (Venkatesan et al., 1997; Silverman and Dracup, 2000; 49


3

Toth et al., 2000; Pongracz et al., 2001; Sivapragasam et al., 2001; Brath et al., 2002; Lin and 50

Chen, 2005; Chattopadhyay and Chattopadhyay, 2007; Guhathakurta, 2008; Lin et al., 2009). 51

Venkatesan et al. (1997) employed ANN to predict all India summer monsoon rainfall with 52

different meteorological parameters as model inputs. Toth et al. (2000) applied three data-53

driven models, auto-regressive moving average, ANN and K-NN, to short-term rainfall 54

predictions. Results showed that ANN performed the best in terms of the accuracy of runoff 55

forecasting when the predicted rainfalls by the three models were used as inputs of a rainfall-56

runoff model. Pongracz et al. (2001) applied fuzzy inference to monthly rainfall prediction. 57

Chattopadhyay and Chattopadhyay (2007) constructed an ANN model to predict monsoon 58

rainfall in India depending on the rainfall series alone. 59

Recently, the concept of coupling different models has attracted more attention in 60

hydrologic forecasting. They can be broadly categorized into ensemble models and modular 61

(or hybrid) models. The basic idea behind ensemble models is to build several different or 62

similar models for the same process and to integrate them together (Shamseldin et al., 1997; 63

Shamseldin and O’Connor, 1999; Xiong et al., 2001; Abrahart and See, 2002; Kim et al, 64

2006). For example, Xiong et al. (2001) used a Takagi-Sugeno-Kang fuzzy technique to 65

couple several conceptual rainfall-runoff models. Coulibaly et al. (2005) employed an 66

improved weighted-average method to coalesce forecasted daily reservoir inflows from K-67

NN, conceptual model and ANN. Kim et al. (2006) investigated five ensemble methods for 68

improving stream flow prediction. 69

Physical processes in rainfall and/or runoff are generally composed of a number of 70

sub-processes. Their accurate modeling by building of a single global model is sometimes not 71

possible (Solomatine and Ostfeld, 2008). Modular models were therefore proposed where 72


4

sub-processes were first of all identified and then separate models (also called local or expert 73

model) were established for each of them (Solomatine and Ostfeld, 2008). Different modular 74

models were proposed depending on the soft or hard splitting of training data. Soft splitting 75

means the dataset can be overlapped and the overall forecasting output is the weighted-76

average of each local model (Zhang and Govindaraju, 2000; Shrestha and Solomatine, 2006; 77

Wu et al., 2008). Zhang and Govindaraju (2000) examined the performance of modular 78

networks in predicting monthly discharges based on the Bayesian concept. Wu et al. (2008) 79

employed a distributed SVR for daily river stage prediction. On the contrary, there is no 80

overlap of data in the hard splitting and the final forecasting output is explicitly from only 81

one of local models (See and Openshaw, 2000; Hu et al., 2001; Solomatine and Xue, 2004; 82

Sivapragasam and Liong, 2005; Jain and Srinivasulu, 2006; Wang et al., 2006; Corzo and 83

Solomatine, 2007; Lin and Wu, 2009). Hu et al. (2001) developed a range-dependent network 84

which employs a number of multilayer perceptron neural networks to model the river flow in 85

different flow bands of magnitude (e.g. high, medium and low). Their results indicated that 86

the range-dependent network performed better than the conventional global ANN. 87

Solomatine and Xue (2004) used M5 model trees and neural networks in a flood-forecasting 88

problem. Sivapragasam and Liong (2005) divided the flow range into three regions, and 89

employed different SVR models to predict daily flows in high, medium and low regions. 90

Wang et al. (2006) used a crisp modular ANN to make soft or crisp predictions for validation 91

data where each local network was trained using the subsets achieved by either a threshold 92

discharge value or a clustering of input spaces. Lin and Wu (2009) proposed a hybrid ANN 93

model for event-based hourly rainfall prediction where self-organizing map networks are 94


5

used for data cluster analysis and multilayer conceptron networks are employed to serve each 95

cluster to construct mapping between input and output. 96

A hydrological time series can be actually regarded as an integration of stochastic (or 97

random) and deterministic components (Salas et al., 1985). Once the stochastic (noise) 98

component is appropriately eliminated, the deterministic component can then be easily 99

modeled. For the purpose of cleaning hydrological series, many data preprocessing 100

techniques, including Principal component analysis (PCA), wavelet analysis (WA), and 101

singular spectrum analysis (SSA), have been employed in hydrology field by researchers 102

(Sivapragasam et al., 2001; Marques et al., 2006; Hu et al., 2007; Partal and Kişi, 2007; 103

Sivapragasam et al., 2007;Wu et al., 2009). Hu et al. (2007) employed PCA as an input data 104

preprocessing tool to improve the prediction accuracy of the rainfall-runoff neural network 105

models. The use of WA to improve rainfall forecasting was conducted by Partal and Kişi 106

(2007). Their results indicated that WA was promising. SSA has also been recognized as an 107

efficient preprocessing algorithm to avoid the effect of discontinuous or intermittent signals, 108

coupled with neural networks (or similar approaches) for time series forecasting (Lisi et al., 109

1995; Sivapragasam et al., 2001; Baratta et al., 2003). For example, Lisi et al. (1995) applied 110

SSA to extract the significant components in their study on southern oscillation index time 111

series and used ANN for prediction. They reconstructed the original series by summing up 112

the first “p” significant components. Sivapragasam et al. (2001) proposed a hybrid model of 113

support vector machine (SVM) and SSA for rainfall and runoff predictions. The hybrid 114

model resulted in a considerable improvement in the model performance in comparison with 115

the original SVM model. A comparison between WA and SSA in Wu et al. (2009) indicated 116

that SSA performed better than WA. In addition, moving average (MA) is used for data 117


6

preprocessing to improve the performance of ANN by de Vos and Rientjes (2005). They 118

argued that one of reasons on lagged predictions of ANN was due to the use of previous 119

observed data as ANN inputs and suggested that an effective solution was to obtain new 120

model inputs by MA over the original data series. 121

In this paper, one of the main purposes is to develop a modular ANN (MANN) 122

coupled with appropriate data-preprocessing techniques to improve the accuracy of rainfall 123

forecasting. MANN consists of three local models which are associated with three subsets 124

clustered by the fuzzy C-means (FCM) clustering method. To evaluate MANN, LR, K-NN 125

and ANN are employed for comparison. ANN is first used to choose the best model inputs by 126

seven candidate model inputs techniques. Once all forecasting models are established, three 127

data-preprocessing methods (i.e., MA, PCA, and SSA) can be examined. To ensure wider 128

application of the conclusions, four cases consisting of two monthly rainfall series and two 129

daily rainfall series from India and China, are investigated. The remaining part is structured 130

as follows. Methodology is detailed in Section 2 where case studies are first described, and 131

then data-preprocessing techniques and forecasting models are introduced. Section 3 presents 132

modeling methods and their applications to four rainfall series. The optimal model input 133

method and the best data preprocessing can be identified. In Section 4, principal results are 134

shown along with relevant discussions. The last section presents main conclusions. 135

2. Methodology 136

2.1 Study Area and Data 137

Two daily mean rainfall series from Daning and Zhenshui river basins of China, and 138

two monthly mean rainfall series from India and Zhongxian of China, are analyzed. 139


7

The Daning River, a first-order tributary of the Yangtze River, is located at the 140

northeastern side of Chongqing city. The daily rainfall data from Jan. 1, 1988 to Dec. 31, 141

2007 were measured at six raingauges located at the upstream of the study basin (Figure 1). 142

The upstream part is controlled by Wuxi hydrology station, with a drainage area of around 2 143

000 km2. The mean areal rainfall series is calculated by the Thiessen polygon method 144

(hereafter the averaged rainfall series is referred to as Wuxi). 145

The Zhenshui basin is located at the northern side of Guangdong Province and 146

adjoined by Hunan Province and Jianxi Province. The basin belongs to a second-order 147

tributary of the Pearl River and has an area of 7,554 km2. The daily rainfall time series of 148

Zhenwan raingauge was collected between January 1, 1989 and December 31, 1998 149

(hereafter the averaged rainfall series is referred to as Zhenwan). 150

The all Indian average monthly rainfall is estimated from area-weighted observations 151

at 306 land stations uniformly distributed over India. The data, with period spanning from 152

January 1871 to December 2007, are available at the website http://www.tropmet.res.in run 153

by the Indian Institute of Tropical Meteorology. 154

The other monthly rainfall series is from Zhongxian raingauge which is located at 155

Chongqing city, China. The catchment containing this raingauge belongs to a first-order 156

tributary of the Yangtze River. The monthly rainfall data were collected from January 1956 157

to December 2007. 158

Figure 2 shows hyetographs of four rainfall series. A linear fit to each hyetograph is 159

denoted by the dashed line. All series appear stationary at least in a weak sense since these 160

linear fits are close to horizontal. 161


8

In this study, each of data series is partitioned into three parts as training set, cross-162

validation set and testing set. The training set serves the model training and the testing set is 163

used to evaluate the performances of models. The cross-validation set has dual functions: one 164

is to implement an early stopping approach in order to avoid overfitting of the training data 165

and another is to select some best predictions from a number of ANN’s runs. In the present 166

study, 10 best predictions are selected from a total of 20 ANN’s runs. The same data partition 167

is adopted for each series: the first half of the entire data as training set and the first half of 168

the remaining data as cross-validation set and the other half as testing set. 169

Table 1 presents pertinent information about watersheds and some descriptive 170

statistics of the original data and three data subsets, including mean (µ), standard deviation 171

(Sx), coefficient of variation (Cv), skewness coefficient (Cs), minimum (Xmin), and maximum 172

(Xmax). As shown in Table 1, the training set cannot fully include the cross-validation or 173

testing data. Owing to the weak extrapolation ability of ANN, all data are scaled to the 174

interval [-0.9, 0.9] instead of [-1, 1] whilst hyperbolic tangent sigmoid functions are 175

employed as transfer functions in hidden and output layers. 176

2.2 Data preprocessing techniques 177

(1) MA 178

MA smoothes data by replacing each data point with the average of the k 179

neighboring data points, where k may be termed the length of memory window. The method 180

is based on the idea that any large irregular component at any point in time will exert a 181

smaller effect if we average the point with its immediate neighbors (Newbold et al., 2003). 182

The equally weighted MA is the most commonly-used, in which each value of the data 183

carries the same weight in the smoothing process. There are three types of moving modes 184


9

including centering, backward and forward. In a forecasting scenario, only the backward 185

mode is used since the other two modes may necessitate future observed values. For a time 186

series 1 2, , , Nx x x , when the backward moving mode is adopted (Lee et al., 2000), the 187

-k term unweighted moving average *ty is written as 188

1*

0

k

t t iiy y k

(1) 189

where , ,t k N . The choice of the window length k is by a trial and error procedure with 190

a minimization of the loss of the objective function. 191

(2) PCA 192

PCA was first introduced by Pearson (1901) and developed independently by 193

Hotelling (1933), and has now well entrenched as an important technique in data analysis. 194

The central idea is to reduce the dimensionality of a data set consisting of a large number of 195

interrelated variables, while retaining as much as possible of the variation present in the data 196

set. The PCA approach uses all of the original variables to obtain a smaller set of principal 197

components (PCs) which can be used to approximate the original variables. PCs are 198

uncorrelated and are ordered so that the first few retain most of the variation present in the 199

original set. 200

Consider a data matrix X which has n rows (observations) and p column (variables). 201

Let the covariance matrix of X be , where cov( ) ( )TE X X X . The linear transformed 202

orthogonal matrix Z is presented as 203

Z XA (2) 204


10

where Z is the PCs with elements ( ,i j ) of thi observation and thj principal component; A 205

is a ( p p ) matrix with eigenvector elements of the covariance of X , and having 206

T T A A AA I . 207

Because matrix TX X is real and symmetric, it can be expressed as T TX X AΛA 208

where Λ is a diagonal matrix whose nonnegative entries are the eigenvalues (λ , 1, ,i i p ) 209

of TX X . The total variance of the data matrix X is represented as 210

1

trace( ) trace( ) trace( )= λp

ii

TAΛA Λ (3) 211

On the other hand, the covariance matrix of principal components Z is expressed as 212

cov( ) ( ) ( )T T TE E Z Z Z A X XA Λ (4) 213

1

trace( ) trace( )= λp

ii

Z Λ (5) 214

Therefore, the total variance of the data matrix X is identical to the total variance after PCA 215

transformation Z . 216

The solution of PCA, using singular value decomposition (SVD) or determinants of 217

the covariance matrix of X , can provide the eigenvectors A with their 218

eigenvalues, λ , 1, ,i i p , representing the variance of each component after PCA 219

transformation. If the eigenvalues are ordered by 1 2 3λ λ λ λ 0p , the first few PCs 220

can capture most of the variance of the original data while the remaining PCs mainly 221

represent the noise in the data. The percentage of total variance explained by the first thm 222

PCs is 223

1 1

λ λ 100%pm

i ii i

V

(6) 224


11

The higher is the selection of the total data variance, V , the better the properties of the data 225

matrix are preserved. For the sake of the reduction of dimensionality, a small number of PCs 226

are selected, though most of the data variance in selected components still remain. If the 227

transformation is to prevent the collinearity of regression variables, the selected component 228

number m in Eq. (6) can be set for a higher total variance, such as 95% 99%V (Hsu et 229

al., 2002). 230

The original data matrix A can be reconstructed by a reverse operation of Eq. (2) as 231

TX ZA (7) 232

By choosing suitable m ( p ) PCs from Z and accompanying m eigenvectors from A , the 233

original data can be filtered. 234

(3) SSA 235

According to Golyandina et al. (2001), the basic SSA consists of two stages: 236

decomposition and reconstruction. The decomposition stage involves two steps: embedding 237

and SVD; the reconstruction stage also comprises two steps: grouping and diagonal 238

averaging. Consider a real-valued time series 1 2, , , NF x x x of length ( 2)N . Assume 239

that the series is a nonzero series, viz. there exists at least one i such that 0ix . Four steps 240

are briefly presented as follows. 241

1st step: embedding 242

The embedding procedure maps the original time series to a sequence of multi-243

dimensional lagged vectors. Let L be an integer (window length), 1 L N , and be the 244

delayed time at a multiple of the sampling period. The embedding procedure forms 245

( 1)n N L lagged vectors 2 ( 1), , , ,T

i i i i i Lx x x x X , where R Li X , and 246


12

1,2, ,i n . The ‘trajectory matrix’ of the time series is denoted by [ ]1 i n X X XX 247

having lagged vectors as its columns. In other words, the trajectory matrix is 248

1 2 3

1 2 3

1 2 2 2 3 2 2

1 ( 1) 2 ( 1) 3 ( 1)

n

n

n

L L L N

x x x xx x x xx x x x

x x x x

X

(8) 249

If 1 , the matrix X is called Hankel matrix since it has equal elements on the 250

‘diagonals’ where the sum of subscripts of row and column is equal to constant. If 1 , the 251

equal elements in X are not definitely in the ‘diagonals’. 252

2nd step: SVD 253

Let TS XX , 1 2λ ,λ , ,λL denote the eigenvalues of S taken in the decreasing order 254

of magnitude ( 1 2 3λ λ λ λ 0L ) and 1 2, , , LU U U denote the orthonormal system of 255

the eigenvectors of the matrix S corresponding to these eigenvalues. If we denote 256

λTi i i iV UX ( 1, ,i L ) (equivalent to the thi eigenvector of TX X ), then the SVD of the 257

trajectory matrix X can be written as 258

1 L X X X (9) 259

where λ Ti i i i U VX . The matrices iX have rank 1; therefore they are elementary matrices. 260

The collection ( λ , ,i i iU V ) is called the thi eigentriple of the SVD. Note that iU and iV are 261

also the thi left and right singular vectors of X , respectively. 262

3rd step: grouping 263

The purpose of this step is to identify appropriately the trend component, oscillatory 264

components with different periods, and structureless noises by grouping components. This 265


13

step can be skipped if one does not want to precisely extract hidden information by 266

regrouping and filtering of components. 267

The grouping procedure partitions the set of indices {1, , }L into m disjoint subsets 268

1, , mI I , so that the elementary matrix in Eq. (9) is regrouped into m groups. Let 269

1{ , , }pI i i . Then the resultant matrix IX corresponding to the group I is defined as 270

1 pI i i X X X . These matrices are computed for 1, , mI I . By substituting into the 271

expansion (9), one obtains the new expansion 272

1 mII X X X (10) 273

The procedure of choosing the sets 1, , mI I is called the eigentriple grouping. 274

4th step: Diagonal averaging 275

The last step in the basic SSA is the transformation of each resultant matrix of the 276

grouped decomposition (10) into a new series of length N . The diagonal averaging is to find 277

equal elements in the resultant matrix and then to generate a new element by averaging over 278

them. The new element has the same position (or index) as the corresponding elements in the 279

original series. As mentioned in the step 1, the concept of ‘diagonal’ is not true for 1 . 280

Regardless of the value of being larger than or equal to 1, the principle of reconstruction is 281

the same. For 1 , the diagonal averaging can be carried out by the formula recommended 282

by Golyandina et al. (2001). Let Y be a ( L n ) matrix with elements ijy , 1 i L , 1 j n . 283

Let * min( , )L L n , * max( , )n L n and ( 1)N n L . Let *ij ijy y if L n and *

ij jiy y

284

otherwise. Diagonal averaging transfers matrix Y to a series 1 2{ , , , }Ny y y by the following 285

equation 286


14

*

*

*

* *, - 1

1

* * *, - 1*

1

- 1* *

, - 1- 1

11

1

1

- 1

k

m k mm

L

k m k mm

N K

m k mm k K

y for k Lk

y y for L k KL

y for L k NN k

(11) 287

Eq. (11) corresponds to the averaging of the matrix elements over the ‘diagonals’ 1i j k . 288

The diagonal averaging, applied to a resultant matrix IkX , produces a N length series kF , 289

and thus the original series F is decomposed into the sum of m series: 290

1 mF F F (12) 291

As mentioned above, these reconstructed components (RCs) can be associated with the trend, 292

oscillations or noise of the original time series with proper choices of L and the sets of 293

1, , mI I . Certainly, if the third step (namely, grouping) is skipped, F can be decomposed 294

into L RCs. 295

2.3 Forecasting models 296

This section describes four candidate forecasting models. They are LR, K-NN, ANN 297

and MANN. They are usually called data-driven models because they capture the mapping 298

between input (e.g. antecedent rainfall) and output variables (forecasted rainfall) without 299

directly considering the physical laws that underlie the mechanism of rainfall (or 300

precipitation). These models are purely based on the information retrieved from the collected 301

rainfall data. 302

(1) Construction of input/output pairs 303


15

Let 1 2, , , Nx x x stand for a rainfall time series. It can be reconstructed into a series 304

of delay vectors as t 2 ( 1), , , ,t t t t mx x x x X , where t RmX , is the delay time as a 305

multiple of the sampling period and m is the embedded dimension. Suppose that the rainfall 306

( 1)t T mx at T -step lead is related to the vector tX , the available historical data may be 307

summarized into a set of pairs as t ( 1), : 1, ,t T mx t n X , where n stands for the number of 308

pairs, and ( 1)n N m . 309

The functional relationship between the input vector tX at time t and the predicted 310

output ( 1)Ft T mx

at time t T can be written as follows: 311

( 1) ( )Ft T m t tx f e X (13) 312

where te is a typical noise term, ( 1)Ft T mx

is the prediction of ( 1)t T mx , and ( )f is the 313

mapping function. The difference of various data-driven forecasting models used in the 314

current study relies on the way of approximating ( )f once model inputs are attained with 315

the appropriate selection of ( , m ). 316

(2) LR 317

The linear regression model herein is actually called stepwise linear regression (SLR) 318

model because the forward stepwise regression is used to determine optimal input variables. 319

The basic idea of SLR is to start with a function that contains the single best input variable 320

and to subsequently add potential input variables to the function one at a time in an attempt to 321

improve the model performance. The order of addition is determined by using the partial 322

-F test values to select which variable should enter next. The high partial -F value is 323

compared to a (selected or default) -F to-enter value. After a variable has been added, the 324


16

function is examined to see if any variable should be deleted. Interested readers are referred 325

to Draper and Smith (1998) and McCuen (2005) for more details. 326

(3) K-NN 327

The prediction of ( 1)t T mx by the K-NN method is formulated as: 328

( 1) ( 1)( , )

1Ft T m t T m

t S n

x xK

X

(14) 329

where S( , n)X denotes the set of indices t of the K nearest neighbors to the feature vector 330

(n)X . The meaning of “nearest neighbors” is generally interpreted in a Euclidean sense. 331

Therefore, if i belongs to S( , n)X and j is not in S( , n)X , then according to Euclidean 332

distance - -n i n jX X X X . Intuitively speaking, the forecast ( 1)Ft T mx in Eq. (14) is the 333

sample average of output rainfall of the K nearest neighbors to (n)X . Obviously, a key task 334

is to determine the parameter K in the K-NN method. 335

(4) ANN 336

The multilayer perceptron network is by far the most popular ANN paradigm, which 337

usually uses the technique of error back propagation to train the network configuration. The 338

architecture of the ANN consists of a number of hidden layers and a number of neurons in 339

the input layer, hidden layers and output layer. ANNs with one hidden layer are commonly 340

used in hydrologic modeling (Dawson and Wilby, 2001; de Vos and Rientjes, 2005) since 341

these networks are considered to provide enough complexity to accurately simulate the 342

nonlinear-properties of the hydrologic process. Based on Eq. (13), the ANN forecasting 343

model is formulated as 344

( 1) 0 ( 1)1 1

( , , , , ) ( )j

h mF outt T m t ji t i j

j i

x f w m h w w x

X (15) 345


17

where denotes transfer functions; jiw are the weights defining the link between the ith 346

node of the input layer and the jth node of the hidden layer; j are biases associated to the 347

jth node of the hidden layer; j

outw are the weights associated to the connection between the 348

jth node of the hidden layer and the node of the output layer; and 0 is the bias at the output 349

node. To apply Eq. (15) to rainfall predictions, appropriate training algorithm is required to 350

optimize w and . 351

(5) MANN 352

To construct MANN, the training data have to be divided into several clusters 353

according to cluster analysis techniques, and then each single model is applied to each cluster. 354

The FCM clustering technique is adopted in the present study (e.g., Bezdek, 1981, Wang et 355

al., 2006). It is able to generate either soft or crisp clusters. ANN (or similar techniques) is 356

unable to extrapolate beyond the range of the data used for training. Otherwise, poor 357

forecasts or predictions can be expected when a new input data is outside the range of those 358

used for training. Hard forecasting is, therefore, taken into consideration in this study. 359

Figure 3 displays the schematic diagram of MANN where the training data is 360

partitioned into three clusters which are based on an assumption that three magnitudes of 361

rainfall (i.e., low, medium, and high) may be derived from different mechanisms. According 362

to this flow chart, once input-output pairs are obtained, they are first split into three subsets 363

by the FCM technique, and then each subset is approximated by a single ANN. The final 364

output of the modular model results directly from the output of one of three local models. 365

2.4 Implementation framework of rainfall forecasting 366


18

Figure 4 illustrates the implementation framework of rainfall forecasting where four 367

prediction models can be conducted in two modes: without/with three data preprocessing 368

methods (dashed box). These acronyms in the column of “methods for model inputs” 369

represent seven methods to determine model inputs: LCA (linear correlation analysis, 370

Sudheer et al., 2002), AMI (average mutual information, Fraser and Swinney, 1986), PMI 371

(partial mutual information, May et al., 2008), FNN (false nearest neighbors, Kennel et al., 372

1992), CI (correlation integral, Theiler, 1986), SLR, and MOGA (ANN based on multi-373

objective genetic algorithm, Giustolisi and Simeone, 2006). 374

2.5 Evaluation of model performances 375

The Pearson’s correlation coefficient (r) or the coefficient of determination (R2 = r2), 376

have been identified as inappropriate measures in hydrologic model evaluation by Legates 377

and McCabe (1999). The coefficient of efficiency (CE) (Nash and Sutcliffe, 1970) is a good 378

alternative to r or R2 as a “goodness-of-fit” or relative error measure in that it is sensitive to 379

differences in the observed and forecasted means and variances. Legates and McCabe (1999) 380

also suggested that a complete assessment of model performance should include at least one 381

absolute error measure (e.g., root mean square error (RMSE)) as necessary supplement to a 382

relative error measure. Besides, the Persistence Index (PI) (Kitanidis And Bras, 1980) was 383

adopted here for the purpose of checking the prediction lag effect. Three measures are 384

therefore used in this study. They are listed below. 385

2 2

1 1

ˆCE 1- ( - ) / ( - )n n

i i ii i

y y y y

(16) 386

2

1

1ˆRMSE ( - )

n

i iiy y

n (17) 387


19

2 2-

1 1

ˆPI 1- ( - ) / ( - )n n

i i i i li i

y y y y

(18) 388

In these equations, n is the number of observations, ˆiy stands for the forecasted flow, 389

iy represents the observed flow, y denotes the average observed flow, and i ly is the flow 390

estimate from a so-call persistence model (or termed naïve model) that basically takes the last 391

flow observation (at time i minus the lead time l ) as a prediction. CE and PI values of 1 392

stands for perfect fits. A small value of PI may imply occurrence of lagged prediction. 393

3. Applications of Models 394

3.1 Determination of model inputs 395

ANN, equipped with the Levernberg-Marquardt (L-M) training algorithm and 396

hyperbolic tangent sigmoid transfer functions, is used as the benchmark model to examine 397

aforementioned seven model input methods in terms of RMSE. Depending on the simplified 398

algorithm from Yu et al. (2000) (downloaded at http://small.eie.polyu.edu.hk/), the four 399

rainfall series are identified as non-chaotic since the correlation dimension does not display 400

the property of convergence, in particular, for daily rainfall series. Results from remaining six 401

methods are presented in Table 2. These results are based on one step lead prediction and let 402

t+1X be the target value at one-step prediction horizon. It can be seen from RMSE that most 403

of these methods tend to be mutually alternative because their RMSE are close. Owing to the 404

convenience of operation, the LCA method is preferred in this study. Furthermore, Figure 5 405

shows identification of effective inputs in Table 2 for the LCA method. Taking Wuxi and 406

Zhenwan as examples, model inputs should take the previous 5-day and 7-day rainfalls for 407


20

them respectively because the partial auto-correlation function (PACF) value decays within 408

the confidence band around these time lags. 409

3.2 Identification of models 410

The model identification is to determine the structure of a candidate model by using 411

training data to optimize relevant parameters of model control once model inputs have been 412

obtained. The LR model is built by the SLR technique. In terms of one step prediction 413

(viz., 1T ), input variables can be found in Table 2. For example, the LR model for Wuxi 414

can be expressed as 415

1 1 2 4 7 110.421 -0.043 0.044 0.025 0.036 0.03Ft t t t t t tx x x x x x x (19) 416

With respect to K-NN, the model identification consists in finding the optimal K if the 417

m dimensional input vector is determined. Sugihara and Mary (1990) suggested that the 418

value of K was taken as 1K m . On the other hand, the choice of K should ensure the 419

reliability of the forecasting (Fraser and Swinney, 1986). The check of robustness of 420

1K m in terms of RMSE is presented in Figure 6, where K is in the interval of [2, 40]. 421

Adopting the value of K as 1m seems reasonable for the current study because the 422

difference between its RMSE and the minimum RMSE is only 2.9% for Wuxi, 2.9% for 423

Zhenwan, 2.6% for India, and 2.0% for Zhongxian, respectively. Consequently, the value of 424

K is 6 for Wuxi ( 5m ), 8 for Zhenwan ( 7m ), 13 for India ( 12m ), and 14 for 425

Zhongxian ( 13m ), respectively. 426

Based on Eq. (14), the formula for one-step lead prediction in the context of K-NN 427

can be defined as 428

1 11

1i

Ft t

K

iKx x

(20) 429


21

where 1Xit

stands for an observed value associated with a neighbor of the current state. For a 430

T step lead prediction, Eq. (20) becomes 431

1

1i

Ft T t T

K

i

x xK

(21) 432

The identification of ANN structure is to optimize the number of hidden nodes h in 433

the hidden layer with the known model inputs and output. The optimal size h of the hidden 434

layer is found by systematically increasing the number of hidden neurons from 1 to 10 until 435

the network performance on the cross-validation set no longer improves significantly. Based 436

on the L-M training algorithm and hyperbolic tangent transfer functions, the identified 437

configurations of ANN are 5-5-1 for Wuxi, 7-4-1 for Zhenwan, 12-5-1 for India, and 13-3-1 438

for Zhongxian, respectively. The same method is used to identify the structure of MANN, and 439

the only difference is that the identification is repeated three times, with each time being for a 440

local ANN. Consequently, MANN is obtained as 5-5/7/9-1 for Wuxi, 7-4/8/4-1 for Zhenwan, 441

12-3/2/5-1 for India, and 13-1/1/1-1 for Zhongxian, respectively. 442

It is worthwhile to notice that the standardization/normalization of the training data is 443

very crucial in the improvement of the model performance. Two methods can be found in the 444

literature (Dawson and Wilby, 2001; Cannas et al., 2002; Rajurkar et al, 2002; Campolo et al., 445

2003; Wang et al., 2006). The standardization (also termed rescaling in some papers) method, 446

as adopted above for model input determination, is to rescale the training data to [-1, 1], [0, 1] 447

or even more narrow interval depending on what kinds of transfer functions are employed in 448

ANN. The normalization method is to rescale the training data to a Gaussian function with a 449

mean of 0 and unit standard deviation, which is by subtracting the mean and dividing by the 450

standard deviation. When the normalization approach is adopted, ANN uses the linear 451

function (e.g. purelin) instead of the hyperbolic tangent sigmoid transfer function in the 452


22

output layer. In addition, some studies have indicated that considerations of statistical 453

principles may improve ANN model performance (e.g. Cheng and Titterington, 1994). For 454

example, the training data was recommended to be normally distributed (Fortin et al., 1997). 455

Sudheer et al. (2002) suggested that the issue of stationarity should be considered in the ANN 456

development because the ANN cannot account for trends and heteroscedasticity in the data. 457

Their results showed that data transformation to reduce the skewness of data was capable of 458

significantly improving the model performance. For the purpose of obtaining better model 459

performance, four data-transformed schemes are examined: 460

Standardizing the raw data (referred to as Std_raw); 461

Normalizing the raw data (referred to as Norm_raw); 462

Standardizing the n-th root transformed data (referred to as Std_nth_root); 463

Normalizing the n-th root transformed data (referred to as Norm_nth_root). 464

Table 3 compares the ANN model performance of the four schemes in terms of 465

RMSE and CE. The Norm_raw scheme is, on the whole, slightly more effective than the 466

Std_raw method. It can also be seen that the effect of the n-th root scheme (3 is taken after 467

trial and error) on the improvement of the performance is basically negligible. Therefore, the 468

Norm_raw scheme is adopted for the later rainfall prediction in the present study. 469

3.3 Rainfall data preprocessing 470

(1) MA 471

The MA operation entails the window length k in Eq. (1) to smooth the raw rainfall 472

data. An appropriate k can be found by systematically increasing k from 1 to 10. The 473

smoothed data is then used to feed into each forecasting model. The targeted value of k 474

corresponds to the optimal model performance in terms of RMSE. 475


23

(2) PCA 476

PCA is employed in two ways: one for reduction of the dimensionality or preventing 477

collinearity (depending on Eq. (2)); second for noise reduction by choosing leading 478

components (contributing most of the variance of the original rainfall data) to reconstruct 479

rainfall series (depending on Eq. (7)). The percentage V of total variance (see Eq. (6)) is set 480

at three horizons, 85%, 90%, and 95% for principal component selection. 481

(3) SSA 482

This approach of filtering a time series to retain desired modes of variability is based 483

on the idea that the predictability of a system can be improved by forecasting the important 484

oscillations in time series taken from the system. The general procedure is to filter the 485

original record first and then to build the forecasting model based on the filtered series. To 486

filter the raw rainfall series, the series needs to be decomposed into components with the aid 487

of SSA. The decomposition by SSA requires identifying the parameter pair ( , L ). The value 488

of an appropriate L should be able to clearly resolve different oscillations hidden in the 489

original signal. However, the present study does not require accurately resolving the raw 490

rainfall signal into trends, oscillations, and noises. A rough resolution can be adequate for the 491

separation of signals and noises where some leading eigenvalues should be identified. 492

To select L , a small interval of [3, 10] is examined in the present study. Figure 7 493

shows the relation between singular spectrum (namely, a set of singular values) and singular 494

number L for Wuxi, Zhenwan, India, and Zhongxian. It can be observed that the curve of 495

singular values in each case except for Wuxi tends to level off with the increase of L . 496

Generally, extraction of high-frequency oscillations becomes more difficult with the increase 497

of singular number L (or mode). L is selected empirically by following the criterion that the 498


24

singular spectrum can be distinguished markedly under that L . According to this criterion, 499

L is set the value of 7 for India and Zhenwan, 6 for Zhongxian. For Wuxi, since all values in 500

the interval satisfy the criterion, in order to reduce computational load in later filtering 501

operation, L is set a small value of 5. The singular spectrum associated with the selected L is 502

highlighted by the dotted solid line in Figure 7. 503

As regards , Figure 8 presents the results of sensitivity analysis of singular spectrum 504

on the lag time using SSA with the determined L . For daily rainfall series, the singular 505

spectrum can be distinguished only when 1 . In contrast, the singular spectrum is 506

insensitive to in the case of monthly rainfall series. The final parameter pair ( , L ) in SSA 507

are set as (1, 5) for Wuxi, (1, 7) for Zhenwan, (1, 7) for India, (1, 6) for Zhongxian, 508

respectively. 509

3.4 Filtering of RCs 510

The subsequent task is to reconstruct a new rainfall series as model inputs by finding 511

contributing RCs so as to improve the predictability of the rainfall series. There is no 512

practical guide on how to identify a contributing or noncontributing component to the 513

improvement of accuracy of prediction. Two proposed filtering methods, supervised and 514

unsupervised, are herein examined. 515

(1) Supervised filtering (denoted by SSA1) 516

Figure 9 depicts cross-correlation function (CCF) between RCs and the original 517

Zhenwan rainfall series. The last plot in this figure presents the average of CCFs from all 7 518

RCs. The average indicates an overall correlation between input and output at various lags 519

(also termed prediction horizons). The plot of average CCF shows that the best correlation is 520

positive and occurs at lag 1. Among all 7 RCs, RC1 exhibits the best positive correlation with 521


25

the original rainfall series. The CCF values for other RCs change alternatively between 522

positive and negative with the increase of the lag. From the perspective of linear correlation, 523

the positive or negative CCF value may indicate that the RC makes a positive or negative 524

contribution to the output of model when the RC is used as the input of model. With the 525

assumption, deleting RCs, which have negative correlations with the model output if the 526

average CCF is positive, may improve the performance of the forecast model. This is the 527

basic idea behind the supervised method. 528

The procedure of the supervised method coupled with ANN is depicted in Figure 10. 529

The aim is to find the optimal p ( L ) RCs from all L RCs for each prediction horizon. The 530

procedure can be summarized into three steps: SSA decomposition, correlation coefficients 531

sorting, and reconstructed components filtering. Operation in each step is bounded by the 532

dashed box. It is worth noting that the filtering method is based on assumption that 533

combination of components with the same sign in CCF ( or -) can strengthen the correlation 534

with the model output. 535

(2) Unsupervised filtering (denoted by SSA2) 536

There are some drawbacks on the supervised method. The salient one is that this 537

method relies on linear correlation analysis, which disregards the existence of nonlinearity in 538

meteorological processes. Also, random combinations among all RCs are not taken into 539

account. To overcome these drawbacks, an unsupervised filtering method (also termed 540

enumeration) is recommended where all input combinations are examined. There are 541

2L combinations for L RCs. The unsupervised method may be computationally intensive if 542

L is large. 543


26

4. Results and Discussions 544

This section presents predictions using various models under two types of modes, 545

namely “normal” and “data preprocessing”. The “data preprocessing” mode is separately 546

described by MA, PCA, and SSA. To extend one-step-ahead prediction to multi-step-ahead 547

prediction, a direct multi-step prediction method (by directly having the multi-step-ahead 548

prediction as output, also termed static prediction method) is adopted in this study to perform 549

two- and three-step-ahead predictions. 550

4.1 Forecasting with normal mode 551

Table 4 shows results of three prediction horizons by applying five models including 552

naïve model to each case study. The naïve model is used as the benchmark in which the 553

forecasted value is directly equal to the last observed value (namely, no change). The naïve 554

model presents the poorest forecasting which can be explained by the fact that it is unlikely to 555

capture any dependence relation. From the perspective of rainfall series, the monthly rainfall 556

can be better predicted than the daily rainfall. Generally, a daily rainfall series, in particular 557

in a semi-humid and semi-dry or dry region, tends to be intermittent and discontinuous due to 558

a large number of no rain periods (dry periods). Two global modeling methods, LR and ANN, 559

mainly capture the zero-zero (or similar extreme low-intensity) rainfall patterns in daily 560

rainfall series because the type of pattern is overwhelmingly dominant in the daily rainfall 561

series. As a consequence, poor performance indices in terms of RMSE, CE, and PI can be 562

observed (depicted in Table 4 for Wuxi and Zhenwan). Nevertheless, Table 4 also shows that 563

MANN performs the best in each case study. MANN adopts three local ANN models, one for 564

each cluster generated by FCM, which can better capture the mapping relation than using a 565

single global ANN. It can be noticed that MANN is more effective for daily rainfall series 566


27

than monthly rainfall data, which can be because daily rainfall data is more irregular (or non-567

periodic) than monthly rainfall series. The use of K-NN for daily rainfall forecasting is even 568

worse than LR although it employs a local prediction approach. Apart from the issue of the 569

selection of K , the performance of K-NN is also influenced by the similarity of input-output 570

patterns. The smooth monthly rainfall series easily construct similar patterns so that they are 571

well predicted by K-NN. It is worth noting that negative values occasionally appear in the 572

forecasts of ANN or MANN whereas this situation does not happen in the K-NN method. 573

Take Wuxi and India data as representative examples, Figure 11 shows the scatter 574

plots and hyetographs of the results at one-day-ahead prediction of ANN and MANN using 575

the rainfall data of Wuxi, where the hyetograph is plotted in a selected range for better visual 576

inspection. ANN seriously underestimates a number of moderate- and high-intensity rainfalls. 577

The low values of CE and PI demonstrate that time shift between the forecasted and observed 578

rainfall may occur, which is further verified by the hyetograph. MANN improves noticeably 579

the accuracy of forecasting in terms of CE and PI. As shown by the scatter plots, the 580

medium-intensity rainfall can be simulated better by MANN although high-intensity rainfalls 581

(or peak values) are still underestimated. Figure 12 shows scatter plots and hyetographs of 582

results at one-day-ahead prediction of ANN and MANN using the rainfall data of India. It 583

can be seen from hyetograph graphs that both ANN and MANN reproduce well the 584

corresponding observed rainfall data, which is further revealed by the scatter plots with a low 585

dispersion around the exact fit line. 586

Figure 13 shows the analysis of the lag effect between forecasted and observed 587

rainfall series. The value of CCF at zero lag corresponds to the actual performance (i.e. 588

correlation coefficient) of the model. A target lag is associated with the maximum value of 589


28

CCF, and is an expression for the mean lag for the forecast. It can be seen from Figure 16 590

that ANN makes fairly obvious lagged predictions for daily rainfall series, and the lag effect 591

can be overcome by MANN. There are 1, 2, and 3 days lag for Wuxi, which are respectively 592

associated with one-, two-, and three-day-ahead forecasting. In contrast, there is no lag effect 593

in monthly rainfall predictions of ANN or MANN. 594

4.2 Forecasting with MA 595

Table 5 presents forecasted results of ANN with the “backward” MA (hereafter 596

referred to as ANN-MA) using the Wuxi rainfall data. The performance indices 597

corresponding to 1k are associated with the normal ANN. Results at each prediction 598

horizon seem to be insensitive to the window length k in view of slight differences among 599

each performance index for k from 1 to 10. Considering the fact that ANNs tend to generate 600

unstable outputs, the influence of MA on the performance of ANN is negligible. Small values 601

of PI also imply that MA cannot eliminate the lagged forecast from ANN. 602

4.3 Forecasting with PCA 603

As mentioned previously, PCA is used in two ways: one (denoted by PCA1) for 604

reduction of dimensionality (also termed principal component regression) and the other one 605

(denoted by PCA2) for noise reduction. Results from PCA1 are presented in Table 6. The 606

scenario of V 100% stands for forecasting using models with the normal mode. Results 607

show that PCA1 cannot improve the model performances in terms of RMSE, CE, and PI, 608

which means that the reduction of dimensionality is unnecessary for the present case studies. 609

Actually, the original inputs are characterized by a low dimension. Table 7 describes the 610

results from PCA2. According to results from LR and K-NN (because results from ANN tend 611

to be unstable), a marginal improvement in the model performances can be observed for the 612


29

Wuxi watershed whereas the model performances deteriorate for the India watershed with the 613

decrease of the value of V . 614

4.4 Forecasting with SSA 615

Following the procedure in Figure 10, the supervised filtering (SSA1) using ANN for 616

RCs of Wuxi and India is illustrated in Figure 14. The RMSE associated with the maximum 617

number of p (for instance 5p for Wuxi) represents the performance of ANN with the 618

normal mode. The optimal p corresponds to the minimum RMSE, which can be found by 619

systematically deleting RCs one at a time. Consequently, numbers of chosen optimal p RCs 620

in three forecasting horizons are 3, 2, and 1 for Wuxi, and 1, 3, and 5 for India, respectively. 621

However, the unsupervised filtering method (SSA2) is based on enumeration of combinations 622

of all RCs. Selection of the optimal p RCs cannot be presented in a graphical form. 623

Table 8 shows selected p at various prediction horizons using LR, K-NN, and ANN 624

in conjunction with SSA1 and SSA2. A large amount of information can be extracted from 625

this Table. First of all, a considerable improvement in the model performance is achieved by 626

each forecasting model in conjunction with SSA1 or SSA2, compared with results in Table 4. 627

From the perspective of rainfall series, the accuracy of daily rainfall prediction is improved 628

significantly in comparison to that in the normal mode. Secondly, as expected, results from 629

SSA2 are superior to or at least equivalent to those from SSA1 since the former examines 630

each combination of RCs in search of the optimal p . SSA2 is therefore considered as an 631

efficient and effective method if the number L of RCs is small. For the present four cases, 632

SSA2 method is appropriate due to the small number of RCs. Once L is large, say 40 or 50, 633

SSA1 may be a good alternative where a relative optimal forecasting can be guaranteed. 634


30

Additionally, it should be noted that the optimal p are different at three forecast horizons. 635

Finally, Table 8 also shows that, among the three models, ANN performs the best with SSA1 636

or SSA2, which is consistent with results in the normal mode. 637

In the normal mode, MANN has been proved to be superior to ANN, in particular, for 638

daily rainfall forecasting. As an attempt to improve the accuracy of rainfall forecasting, 639

MANN is also coupled with SSA2. Table 9 demonstrates results in terms of RMSE, CE, and 640

PI using MANN compared with those of ANN. Good accuracies of forecasting are made by 641

both MANN and ANN. It can be seen from values of PI that the prediction lag effect is 642

completely eliminated. The model performance does not deteriorate markedly with the 643

increase of the forecasting lead. Results also show that MANN still maintains a salient 644

superiority over ANN in the SSA2 mode for both daily and month rainfall series. 645

One-step lead estimates of MANN and ANN with the help of SSA2 are shown in 646

Figure 15 (Wuxi) and Figure 16 (India) in the form of hyetographs and scatter plots (the 647

former is plotted in a selected range for better visual inspection). Compared with Figure 11, 648

each scatter plot in Figure 15 is closer to the exact line, which means that the daily rainfall 649

process is fitted appropriately. Nevertheless, some peak values still remain mismatched 650

although MANN shows a better ability to capture the peak value than ANN. Regarding the 651

monthly rainfall series, the scatter plots with perfect match of the diagonal indicates that the 652

rainfall process is perfectly reproduced. The representative hyetograph shows that the peak 653

times and peak values are also accurately predicted. 654

Figure 17 presents the correlation analysis between observed and forecasted rainfall 655

from ANN and MANN using the Wuxi and India series, respectively. Compared with Figure 656


31

13, the lagged prediction of ANN is completely eliminated by SSA2 since the maximum 657

CCF occurs at zero lag. The larger the CCF at zero lag is, the better the model performance is. 658

4.5 Discussions 659

Some discussions regarding forecasting models and the effects of the SSA technique 660

are made in the following. 661

(1) About the investigation of effects of SSA 662

Figure 18 shows that a large number of zeros and near zeros occur in the original 663

Wuxi rainfall which makes the series discontinuous. Using the intermittent series is difficult 664

to reconstruct similar input patterns for a forecasting model. Thus, depending on those 665

reconstructed input patterns, data-driven models based on pattern training, for example, 666

ANNs, tend to be unfeasible. In contrast, rainfall series preprocessed by SSA becomes 667

smoother where most of zeros are replaced by nonzero values. New input vectors from the 668

reconstructed rainfall series are characterized by better repetition of patterns so that they are 669

easier reproduced. 670

To investigate the influence of SSA on the ANN’s performance, correlation analyses 671

between inputs and output of ANN and ANN-SSA2 are compared using the Wuxi data and 672

are depicted in Figure 19. As the input and output series in ANN are both the raw rainfall 673

series, the cross-correlation analysis is equivalent to the autocorrelation analysis of the raw 674

rainfall series. At all three prediction horizons, cross-correlation coefficients (CCs) between 675

reconstructed inputs by SSA2 and the raw rainfall data are improved significantly at most 676

lags except for the lag of 3 at one-step lead. It should be recalled that model inputs are the 677

previous five rainfall data for Wuxi. The “starting point” in Figure 19 represents the first 678

previous rainfall of the five inputs, and the remaining inputs consist of four points after the 679


32

starting point. It can be observed that the CCF value between each new model input and 680

output are far larger than that between the raw model input and output (seen at the same lag). 681

Therefore, the improvement of a model’s performance by the SSA technique may be owing 682

to the enhancement of the mapping relation of model input and output by deleting noises 683

hidden in the raw signals. 684

(2) About parameter L in SSA 685

The parameter L in SSA has a significant impact on the performance of a forecasting 686

model since the optimal p RCs may be different with the change of L when using the same 687

forecasting model at the same prediction horizon. The selection of L in this study is based on 688

the interval of [3, 10] in conjunction with an empirical criterion (namely, a particular L is 689

selected only if the singular spectrum can be distinguished markedly under that L ). To check 690

the robustness of the empirical method, each L in [3, 10] is examined by the LR model with 691

SSA2 using the Wuxi and Zhenwan data and presented in Table 10. As mentioned previously, 692

the target L for Wuxi and Zhenwan are 5 and 7, respectively. The RMSE associated with 693

them at each prediction horizon is highlighted in bold (shown in Table 10). In terms of Wuxi, 694

the difference between the target RMSE and the minimum RMSE at the same prediction 695

horizon is only 9.2% for one-step prediction, 1.4% for two-step prediction, 0.0% for three-696

step prediction, respectively. Regarding Zhenwan, the three values are respectively 5.2%, 697

7.5%, and 0.0%. These changes are slight and cannot influence the conclusions drawn 698

previously. Therefore, the empirical method for the present rainfall data should be 699

appropriate. 700


33

5. Conclusion 701

This study suggests the use of modular artificial neural network (MANN) coupled 702

with data preprocessing techniques for improving four rainfall predictions from India and 703

China consisting of two monthly and two daily series. To reasonably evaluate MANN’s 704

performance, three models, LR, K-NN and ANN, are used for the purpose of comparison. In 705

the process of model development, model inputs and data preprocessing techniques are 706

carefully analyzed and discussed. The following conclusions are reached based on this study: 707

1. LCA is regarded as an effective and efficient method among all seven input 708

techniques due to its simplicity of computation and comparable capability of forecasting. 709

2. In the normal mode (without data preprocessing), MANN distinguishes from the 710

other three models for both monthly and daily rainfall series forecasting. Whilst all four 711

models reasonably forecast two monthly rainfall series, only MANN is able to simulate each 712

daily rainfall series without obvious lag effect. 713

3. In the data preprocessing mode, the effect of MA is negligible for the improvement 714

of each forecasting model. 715

4. PCA as a data preprocessing technique is discussed in two forms, i.e. PCA1 for the 716

purpose of dimension reduction, and PCA2 for the purpose of noise reduction. Results show 717

that PCA1 cannot improve model’s performance and PCA2 marginally improve model’s 718

performance. 719

5. Two filtering method, i.e. supervised (SSA1) and unsupervised (SSA2), are 720

examined for SSA when coupled with forecasting models. It can be found that each model 721

achieves considerable improvement in performance with the aid of SSA1 or SSA2. In terms 722

of forecasting models, MANN still outperforms all other models. 723


34

6. As far as two filtering methods are concerned, SSA2 tends to be better if the 724

number of raw RCs is small. Otherwise, SSA1 is a good alternative. 725

7. A further discussion reveals that the essence of SSA in improving model 726

performance is to strengthen the mapping relation of model input and output by deleting 727

noises in the raw signal. 728

8. There is still considerable room for improving forecasting of peak values although 729

MANN coupled with SSA has made perfect overall predictions for daily rainfall series. 730

731

Nomenclature 732

ACF Auto Correlation Function 733

AMI Average Mutual Information 734

ANN Artificial Neural Networks 735

CC Cross-correlation Coefficient 736

CCF Cross Correlation Function 737

CE Coefficient of Efficiency 738

CI Correlation Integral 739

FCM Fuzzy C-means 740

FNN False Nearest Neighbor 741

K-NN K-Nearest-Neighbors 742

LCA Linear Correlation Analysis 743

L-M Levenberg-Marquart 744

LR Linear Regression 745

MA Moving Average 746


35

MANN Modular Artificial Neural Networks 747

MOGA ANN based on Multi-objective Genetic Algorithm 748

PACF Partial Auto Correlation Function 749

PC Principal Component 750

PCA Principal Component Analysis 751

PMI Partial Mutual Information 752

PI Persistence Index 753

RC Reconstructed Component 754

RMSE Root Mean Square Error 755

SLR Stepwise Linear Regression 756

SSA Singular Spectrum Analysis 757

SVD Singular Value Decomposition 758

SVM Support Vector Machine 759

SVR Support Vectors Regression 760

WA Wavelet Analysis 761

762

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ensemble technique based on singular spectrum analysis to daily rainfall forecasting. Neural 768

Networks, 16, 375-387. 769


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945


45

Figure Captions 946

Figure1. Location of Daning river basin (Map of Chongqing in the left panel, and Daning 947

watershed in the dashed box) 948

Figure 2. Rainfall series of (a) Wuxi, (b) India, (c) Zhenwan, and (d) Zhongxian 949

Figure 3. Flow chart of hard forecasting using a modular model 950

Figure 4. Implementation framework of forecast models with/without data preprocessing 951

Figure 5. Plots of ACF and PACF of the rainfall series with the 95% confidence bounds (the 952

dashed lines), (a) and (c) for Wuxi, and (b) and (d) for Zhenwan 953

Figure 6. Check of robustness of K in KNN method for (a) Wuxi, (b) India, (c) Zhenwan, 954

and (d) Zhongxian. 955

Figure 7. Singular Spectrum as a function of lag using various window lengths L for (a) Wuxi, 956

(b) India, (c) Zhenwan, and (d) Zhongxian 957

Figure 8. Sensitivity analysis of singular Spectrum on varied for (a) Wuxi, (b) India, (c) 958

Zhenwan, and (d) Zhongxian 959

Figure 9. Plots of CCF between each RC and the raw rainfall data for Zhenwan 960

Figure 10. Supervised procedure for a forecast model with SSA 961

Figure 3. Scatter plots and hyetographs of one-step-ahead forecast using ANN and MANN 962

for Wuxi ((a) and (c) from ANN, and (b) and (d) from MANN) 963


for India ((a) and (c) from ANN, and (b) and (d) from MANN) 965

Figure 5. CCFs at three forecast horizons for various lags in time of observed and forecasted 966

rainfall of ANN and MANN: Wuxi (left column) and India (right column) 967

Figure 6. Performance of ANN with SSA1 in terms of RMSE as a function of p ( L ) 968


46

selected RCs at various prediction horizons: (a) Wuxi and (b) India 969


with SSA2 for Wuxi ((a) and (c) from ANN, and (b) and (d) from MANN) 971


with SSA2 for India ((a) and (c) from ANN, and (b) and (d) from MANN) 973

Figure 17. CCFs at three forecast horizons using ANN and MANN with Wuxi and India ((a) 974

and (c) for Wuxi and (b) and (d) for India) 975

Figure 18. Hyetographs (detail; from 900th to 950th) of Wuxi: (1) the raw rainfall series and (2) 976

reconstructed rainfall series for one-step prediction of ANN. 977

Figure 19. CCF between inputs and output for ANN and ANN-SSA2 at three prediction 978

horizons using the Wuxi data 979

980


47

Table captions 981

Table 1. Pertinent information for four watersheds and the rainfall data 982

Table 2. Comparison of methods to determine mode inputs using ANN model 983

Table 3. Performance comparison of ANN with different data-transformed methods 984

Table 4. Model performances at three forecasting horizons under normal mode 985

Table 5. Model performances of ANN-MA using the Wuxi data 986

Table 6. Multiple-step predictions for Wuxi and India series using LR, K-NN, and ANN with 987

PCA1 988

Table 7. Multiple-step predictions for Wuxi and India series using LR, K-NN, and ANN 989

with PCA2 990

Table 8. Optimal p RCs for model inputs at various forecasting horizons 991

Table 9. Model performances at three forecasting horizons using MANN and ANN with the 992

SSA2 993

Table 10. RMSE of the LR model coupled with SSA2 using various L 994

995


48

0 15 30 45 60 75 km

Chongqing

Gaolou

Jianlou

Changan

XujiabaXining

Wuxi

Wanzhou

Yunyan

KaixianFengjie

Wushan

YangtzeRi

ver

Gaolou

Jianlou

Changan

XujiabaXining

Wuxi

Wushantown or cityhydrology stationrain gauge

996

Figure 1. 997

998


49

999

1000

1000 2000 3000 4000 5000 6000 70000

50

100

150

200

Time (1/1/1988-31/12/2007)

Rai

nfal

l (m

m/d

ay)

500 1000 1500 2000 2500 3000 35000

50

100

150

200

Time (1/1/1989-31/12/1998)

Rai

nfal

l (m

m/d

ay)

200 400 600 800 1000 1200 1400 16000

1000

2000

3000

4000

Time (1/1871-12/2007)

Rai

nfal

l (m

m/m

onth

)

100 200 300 400 500 6000

200

400

600

Time (1/1956-12/2007)

Rai

nfal

l (m

m/m

onth

)

Wuxi

Linear fitting

Zhenwan

Linear fitting

India

Linear fitting

Zhongxian

Linear fitting

(d)(c)

(b)(a)

1001

Figure 2. 1002

1003


50

1004

Figure 3. 1005

1006

Final output

( 1)Ft T mx

Raw input series

x

FCM clustering

Sub-model 1

Determination of model

inputs

Sub-model 2

Sub-model 3


51

1007

1008

Figure 4. 1009

Raw input series

x

LCA

AMI

PMI

FNN

CI

SLR

MOGA

Methods for model inputs

MA

PCA

SSA

MA

PCA

SSA

LR

K-NN

ANN

MANN

MA

PCA

SSA

MA

PCA

SSA

Forecasting models

Method for data preprocessing

( 1)Ft T mx

Model output

( 1)Ft T mx

( 1)Ft T mx

( 1)Ft T mx


52

1010

1011

0 5 10 15 20-0.2

0.2

0.6

1

AC

F

Wuxi

0 5 10 15 20-0.2

0.2

0.6

1

AC

F

Zhenwan

0 5 10 15 20-0.2

0.2

0.6

1

Lag (day)

PA

CF

0 5 10 15 20-0.2

0.2

0.6

1

Lag (day)

PA

CF

(c) (d)

(a) (b)

1012 Figure 5. 1013

1014


53

1015

10 20 30 4010

11

12

13

K

Val

i-RM

SE

Wuxi

10 20 30 40200

220

240

260

280

300

K

Val

i-RM

SE

India

10 20 30 4010

11

12

13

14

15

K

Val

i-RM

SE

Zhenwan

10 20 30 4052

54

56

58

K

Val

i-RM

SE

Zhongxian(d)(c)

(b)(a)

K=m+1minimum

K=m+1minimum

minimum

minimum

K=m+1

K=m+1

1016

Figure 6. 1017

1018


54

1 3 5 7 96

6.5

7

7.5ln

( s

ingu

lar

valu

e )

Mode

Wuxi

1 3 5 7 96.2

6.4

6.6

6.8

7

7.2

ln (

sin

gula

r va

lue

)

Mode

Zhenwan

1 3 5 7 99

10

11

12

ln (

sin

gula

r va

lue

)

Mode

India

1 3 5 7 97

7.5

8

8.5

9

ln (

sin

gula

r va

lue

)

Mode

Zhongxian

(a) (b)

(c) (d)

L=10

L=3 L=3

L=10

L=3

L=10

L=3

L=10

1019

Figure 7. 1020

1021


55

1022

1023

1024

1 2 3 4 56.3

6.8

7.3

ln (

sin

gula

r va

lue

)

singular number, L

Wuxi

=1=3=5=7=10

1 2 3 4 5 6 7

6.3

6.7

7.1

ln (

sin

gula

r va

lue

)

singular number, L

Zhenwan

=1=3

=5=7=10

1 2 3 4 5 6 79

10

11

12

ln (

sin

gula

r va

lue

)

singular number, L

India

=1=3=5=7=10

1 2 3 4 5 67

8

9

ln (

sin

gula

r va

lue

)

singular number, L

Zhongxian

=1=3=5=7=10

(d)(c)

(b)(a)

1025

Figure 8. 1026


56

1027

0 2 4 6 8 10 120

0.5

1RC1

CC

F

0 2 4 6 8 10 12-1

0

1 RC2

CC

F

0 2 4 6 8 10 12-1

0

1 RC3

CC

F

0 2 4 6 8 10 12-1

0

1 RC4

CC

F

0 2 4 6 8 10 12-1

0

1RC5

CC

F

0 2 4 6 8 10 12-0.5

0

0.5RC6

CC

F

0 2 4 6 8 10 12-0.5

0

0.5RC7

Lag (day)

CC

F

0 2 4 6 8 10 120

0.5

1Average over all RCs

Lag (day)

CC

F

1028 Figure 9. 1029

1030


57

1031

1032

Figure 10. 1033

1034

Choose (τ, L) for singular spectrum analysis on raw rainfall data

SSA Decomposition

Decompose raw rainfall data into L reconstructed components (RCs)

Obtain correlation coefficient matrix by correlation analysis between RC and raw rainfall data at varied lags (or termed prediction horizons) from 1 to T

Compute average correlation coefficient by averaging over L correlation coefficients at each same lag, as shown in the last plot of Figure 9.

For a specified lag (or prediction horizon), all L values of CCFs at this lag are sorted in a ascending (or ascending) order if the average CCF is negative (or positive) when the pruning algorithm for the filter of RCs is adopted

ANN is repeatedly trained and tested by new time series which is obtained by pruning operation on sorted RCs where the number of p varies from L at the beginning to 1 at the end. The optimal p is associated with the best performance of ANN.

Correlation Coefficients Sort

Reconstructed Components Filter


58

1035

0 50 100 1500

50

100

150

Observed

Fore

cast

ed

ANN

900 910 920 930 940 9500

50

100

150

Time (day)

Rai

nfal

l (m

m)

0 50 100 1500

50

100

150

Observed

Fore

cast

ed

MANN

900 910 920 930 940 9500

50

100

150

Time (day)

Rai

nfal

l (m

m)

observedforecasted

observedforecasted

RMSE: 8.2CE: 0.50PI: 0.60

RMSE: 10.6CE: 0.17PI: 0.32

(b)

(d)(c)

(a)

1036 Figure 11. 1037

1038


59

1039

1040

1041

0 500 1000 1500 2000 2500 30000

1000

2000

3000

Observed

Fore

cast

ed

ANN

150 160 170 180 190 2000

1000

2000

3000

4000

Time (month)

Rai

nfal

l (m

m)

0 500 1000 1500 2000 2500 30000

1000

2000

3000

Observed

Fore

cast

ed

MANN

150 160 170 180 190 2000

1000

2000

3000

4000

Time (month)

Rai

nfal

l (m

m)

observedforecasted

observedforecasted

(a)

(c) (d)

(b)

RMSE: 245.2CE: 0.93PI: 0.86

RMSE: 243.3CE: 0.93PI: 0.86

1042 Figure 12. 1043

1044


60

1045

-10 -5 0 5 100

0.5

1

Lag (day)

CC

F

ANN (Wuxi)

-10 -5 0 5 100

0.5

1

Lag (day)

CC

F

MANN (Wuxi)

-10 -5 0 5 10-1

-0.5

0

0.5

1

Lag (month)

CC

F

ANN (India)

-10 -5 0 5 10-1

-0.5

0

0.5

1

Lag (month)

CC

F

MANN (India)

One-day

Two-day

Three-day

One-day

Two-day

Three-day

One-month

Two-month

Three-month

One-month

Two-month

Three-month

(b)

(d)(c)

(a)

1046 Figure 13. 1047

1048


61

1049

1050

1051

5 4 3 2 14

5

6

7

8

9

10

11

12

p

RM

SE

(m

m)

Wuxi

one steptwo stepthree step

7 6 5 4 3 2 1160

180

200

220

240

260

280

300

p

RM

SE

(m

m)

India

one steptwo stepthree step

(a) (b)

1052 Figure 14. 1053

1054


62

0 50 100 1500

50

100

150

Observed

Fore

cast

edANN-SSA2

900 910 920 930 940 9500

50

100

150

Time (day)

Rai

nfal

l (m

m)

0 50 100 1500

50

100

150

Observed

Fore

cast

ed

MANN-SSA2

900 910 920 930 940 9500

50

100

150

Time (day)

Rai

nfal

l (m

m)

observedforecasted

observedforecasted

(a)

(c) (d)

(b)

RMSE: 4.43CE: 0.84PI: 0.87

RMSE: 3.63CE: 0.90PI: 0.92

1055 Figure 15. 1056

1057


63

1058

1059

1060

0 500 1000 1500 2000 2500 30000

1000

2000

3000

Observed

Fore

cast

ed

ANN-SSA2

150 160 170 180 190 2000

1000

2000

3000

4000

Time (month)

Rai

nfal

l (m

m)

0 500 1000 1500 2000 2500 30000

1000

2000

3000

Observed

Fore

cast

ed

ANN-SSA2

150 160 170 180 190 2000

1000

2000

3000

4000

Time (month)

Rai

nfal

l (m

m)

observedforecasted

observedforecasted

RMSE: 144.2CE: 0.98PI: 0.95

RMSE: 164.7CE: 0.98PI: 0.95

(b)

(d)(c)

(a)

1061 Figure 16. 1062

1063


64

1064

1065

1066

-10 -5 0 5 10-0.5

0

0.5

1

Lag (day)

CC

F

ANN-SSA2(Wuxi)

-10 -5 0 5 10-0.5

0

0.5

1

Lag (day)

CC

F

MANN-SSA2(Wuxi)

-10 -5 0 5 10-1

-0.5

0

0.5

1

Lag (month)

CC

F

ANN-SSA2(India)

-10 -5 0 5 10-1

-0.5

0

0.5

1

Lag (month)

CC

F

MANN-SSA2(India)

One-dayTwo-dayThree-day

One-dayTwo-dayThree-day

One-monthTwo-monthThree-month

One-monthTwo-monthThree-month

(a)

(c) (d)

(b)

1067 Figure 17. 1068

1069


65

1070

1071

900 905 910 915 920 925 930 935 940 945 9500

20

40

60

Time (day)

Rai

nfal

l (m

m)

Original Wuxi rainfall series

900 905 910 915 920 925 930 935 940 945 950

0

20

40

60

Time (day)

Rai

nfal

l (m

m)

Reconstructed Wuxi rainfall series

(a)

(b)

1072

Figure 18. 1073

1074


66

1075

0 5 10 150

0.2

0.4

0.6

0.8

1

Lag (day)

CC

F

One-step lead

0 5 10 150

0.2

0.4

0.6

0.8

1

Lag (day)

Two-step lead

0 5 10 150

0.2

0.4

0.6

0.8

1

Lag (day)

Three-step lead

Input series generated by SSA2 Raw input series

starting point

starting point

starting point

1076

Figure 19. 1077

1078


67

Table 11. Pertinent information for four watersheds and the rainfall data 1079

Watershed and datasets

Statistical parameters Watershed area and data period

µ Sx Cv Cs Xmin Xmax (mm) (mm) (mm) (mm)

Wuxi Original data 3.67 10.15 0.36 5.68 0.00 154 Area: Training 3.81 10.94 0.35 6.27 0.00 147 2 000 km2 Cross-validation 3.42 8.87 0.39 4.96 0.00 102 Data period: Testing 4.03 11.60 0.35 5.46 0.00 154 Jan., 1988- Dec., 2007 Zhenwan Original data 4.3 11.0 0.39 4.94 0.0 159 Area: Training 4.3 11.2 0.38 5.60 0.0 159 7554 km2 Cross-validation 4.7 11.2 0.42 4.22 0.0 125 Data period: Testing 4.0 10.9 0.37 4.97 0.0 133 Jan., 1989- Dec., 1998 India Original data 906.7 951.6 1.0 0.9 3.0 3460 Area:

Training 904.8 955.7 0.9 0.9 3.0 3393 all India

Cross-validation 918.2 969.5 0.9 1.0 8.0 3460 Data period:

Testing 898.9 927.4 1.0 0.9 16.0 3232 Jan., 1871- Dec., 2007 Zhongxian Original data 96.2 79.2 1.2 1.2 0.0 599 Area: Training 97.2 77.5 1.3 0.9 0.0 429 Cross-validation 98.6 86.8 1.1 1.9 0.0 599 Data period: Testing 91.8 74.9 1.2 0.8 0.0 306 Jan., 1956- Dec., 2007

1080

1081


68

Table 12. Comparison of methods to determine mode inputs using ANN model 1082

Watershed Methods m Effective inputsa Identified ANN

RMSE

Wuxi

LCA 1 20 The last 5 (5-5-1) 10.74

AMI 1 12 Except for Xt-10,t-9 (10-3-1) 10.91

PMIb 1 12 Xt,t-1,t-3,t-5,t-7,t-10,t-4 (7-8-1) 10.85

FNN 1 20 The last 14 (14-3-1) 11.02

CI 4 20 Nil

SLR 1 12 Xt-11,t-7,t-4,t-2,t-1,t (6-3-1) 10.94

MOGA 1 12 Xt, t-1 (2-6-1) 10.55

Zhenwan

LCA 1 20 The last 7 (7-4-1) 11.03

AMI 1 12 Except for Xt-11,t-10,t-9,t-8,t-2 (7-5-1) 10.95

PMI 1 12 Xt-4,t,t-1,t-3,t-11,t-5,t-10,t-6,t-9 (10-4-1) 10.98

FNN 1 20 Last 14 (14-3-1) 11.08

CI 3 20 Nil

SLR 1 12 Xt-11,t-7,t-6,t-3,t-1,t (6-3-1) 11.01

MOGA 1 12 Xt,t-4,t-7,t-9,t-11 (5-8-1) 10.43

India

LCA 1 20 the last 12 (12-5-1) 256.22

AMI 1 12 the last 12 (12-5-1) 256.22

PMI 1 12 Xt-11,t-10,t-5,t (4-5-1) 275.06

FNN 1 20 the last 5 (5-9-1) 286.04

CI 4 20 nil

SLR 1 12 except for Xt-4 (11-9-1) 258.13

MOGA 1 12 Xt-11,t-9,t-7,t-5,t-4,t-1,t (7-1-1) 277.57

Zhongxian

LCA 1 20 the last 13 (13-3-1) 51.70

AMI 1 12 Xt-11,t-10,t-6,t-5,t-4,t (6-5-1) 54.67

PMI 1 12 Xt-11,t,t-9,t-7,t-7 (5-9-1) 55.39

FNN 1 20 the last 4 (4-7-1) 59.78

CI 3 20 nil

SLR 1 12 Xt-11,t-7,t-6,t-5,t-3,t (6-6-1) 55.47

MOGA 1 12 Xt-11,t-10,t-6,t-3,t (5-2-1) 53.93

Note:a for the convenience of writing down effective inputs, “Xt, t-1”stands for Xt, Xt-1; b effective inputs 1083 from PMI are in descending order of priority. 1084

1085


69

Table 13. Performance comparison of ANN with different data-transformed methods 1086

Watershed Data

Transformation RMSE CE

1 2 3 a 1 2 3 Wuxi Std_raw 10.77 11.54 11.62b 0.14 0.01 0.00 Norm_raw 10.57 11.49 11.59 0.17 0.02 0.00 Std_nth_root 11.00 12.02 12.10 0.10 -0.07 -0.09 Norm_nth_root 11.15 12.01 12.09 0.08 -0.07 -0.09 Zhenwan Std_raw 11.03 11.11 11.16 0.03 0.02 0.01 Norm_raw 10.72 11.06 11.14 0.09 0.03 0.02 Std_nth_root 11.25 11.68 11.75 -0.01 -0.09 -0.10 Norm_nth_root 11.34 11.70 11.74 -0.02 -0.09 -0.09 Wuxi Std_raw 256.22 250.51 249.46 0.92 0.93 0.93 Norm_raw 251.74 246.48 250.99 0.93 0.93 0.93

Std_nth_root 259.81 253.42 256.43 0.92 0.93 0.92

Norm_nth_root 252.75 251.95 259.00 0.93 0.93 0.92

Zhongxian Std_raw 54.26 54.23 53.91 0.48 0.48 0.48 Norm_raw 52.91 53.10 52.78 0.50 0.50 0.51 Std_nth_root 52.15 53.44 53.17 0.52 0.49 0.50 Norm_nth_root 52.27 53.37 54.30 0.51 0.49 0.48

a Numbers of “1, 2, and 3” denote one-, two-, and three-day-ahead forecasting; 1087 b Result is average over 10 best runs from total 20 runs; 1088

1089


70

Table 14. Model performances at three forecasting horizons under the normal mode 1090

Watershed Model RMSE CE PI

1* 2* 3* 1 2 3 1 2 3

WuXi

Naïve 12.2 16.0 16.5 0.05 -0.61 -0.72 0.00 0.00 0.00

LR 10.9 11.9 12.0 0.12 -0.05 -0.07 0.28 0.41 0.43

K-NN 11.8 12.4 12.6 -0.03 -0.14 -0.18 0.16 0.36 0.38

ANN 10.6 11.5 11.6 0.17 0.02 0.00 0.32 0.45 0.47

MANN 8.2 9.2 9.0 0.50 0.38 0.40 0.60 0.65 0.68

Zhenwan

Naïve 12.4 12.2 13.6 -0.53 -0.49 -0.84 0.00 0.00 0.00

LR 11.1 11.3 11.4 0.02 -0.02 -0.03 0.39 0.42 0.45

K-NN 12.7 12.7 12.8 -0.27 -0.28 -0.30 0.21 0.28 0.31

ANN 10.7 11.1 11.1 0.09 0.03 0.02 0.43 0.45 0.48

MANN 7.9 9.6 9.9 0.50 0.27 0.23 0.69 0.59 0.59

India

Naïve 643.1 1084.5 1399.2 0.52 -0.37 -1.28 0.00 0.00 0.00

LR 286.8 301.6 302.7 0.90 0.89 0.89 0.80 0.92 0.95

K-NN 246.6 257.3 251.2 0.93 0.92 0.93 0.85 0.94 0.97

ANN 245.2 245.9 247.2 0.93 0.93 0.93 0.86 0.95 0.97

MANN 243.3 241.8 244.4 0.93 0.93 0.93 0.86 0.95 0.97

Zhongxian

Naïve 75.7 91.9 109.2 -0.03 -0.51 -1.13 0.00 0.00 -0.02

LR 56.1 57.7 58.4 0.44 0.41 0.39 0.46 0.60 0.71

K-NN 55.0 56.0 57.2 0.46 0.44 0.42 0.48 0.63 0.72

ANN 52.5 54.4 54.3 0.51 0.48 0.48 0.52 0.65 0.75

MANN 50.3 50.2 53.2 0.55 0.55 0.50 0.56 0.70 0.76 * The number of “1, 2, and 3” denote one-, two-, and three-step-ahead forecasts 1091

1092


71

1093

Table 15. Model performances of ANN-MA using the Wuxi data 1094

Prediction horizons

Performance index

Window length (k) for MA 1 2 3 4 5 6 7 8 9 10

One-step RMSE 10.60 10.70 10.63 10.72 10.77 10.70 10.83 10.83 10.72 10.68 CE 0.17 0.15 0.16 0.15 0.14 0.15 0.13 0.13 0.15 0.15 PI 0.32 0.31 0.32 0.31 0.30 0.31 0.29 0.29 0.31 0.31 Two-step RMSE 11.50 11.51 11.51 11.50 11.53 11.56 11.55 11.47 11.45 11.45 CE 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.02 0.03 0.03 PI 0.45 0.44 0.44 0.45 0.44 0.44 0.44 0.45 0.45 0.45 Three-step RMSE 11.60 11.58 11.58 11.55 11.61 11.58 11.56 11.51 11.52 11.52 CE 0.00 0.00 0.00 0.01 0.00 0.00 0.01 0.02 0.01 0.01 PI 0.47 0.47 0.47 0.48 0.47 0.47 0.48 0.48 0.48 0.48

1095


72

Table 6. Multiple-step predictions for Wuxi and India series using LR, K-NN, and ANN 1096

with PCA1 1097

Watershed Performance

index V (%)1

LR K-NN ANN 1* 2* 3* 1 2 3 1 2 3

Wuxi RMSE 85 11.0 11.9 12.0 12.1 12.5 12.7 10.6 11.5 11.6 90 11.0 11.9 12.0 12.1 12.5 12.7 10.7 11.5 11.6 95 10.9 11.9 12.0 11.8 12.4 12.6 10.6 11.5 11.6 100 10.9 11.9 12.0 11.8 12.4 12.6 10.6 11.5 11.6 CE 85 0.12 -0.05 -0.07 -0.04 -0.15 -0.19 0.16 0.01 0.00 90 0.12 -0.05 -0.07 -0.04 -0.15 -0.19 0.16 0.02 0.00 95 0.12 -0.05 -0.07 -0.03 -0.14 -0.18 0.16 0.02 0.00 100 0.12 -0.05 -0.07 -0.03 -0.14 -0.18 0.17 0.02 0.00 PI 85 0.27 0.41 0.43 0.12 0.34 0.36 0.32 0.44 0.47 90 0.27 0.41 0.43 0.12 0.34 0.36 0.31 0.45 0.47 95 0.28 0.41 0.43 0.16 0.36 0.38 0.32 0.45 0.47 100 0.28 0.41 0.43 0.16 0.36 0.38 0.32 0.45 0.47 India RMSE 85 410.9 320.6 457.7 275.9 262.7 281.1 250.4 256.1 247.0 90 294.9 307.8 311.1 260.3 256.2 276.8 248.1 252.3 249.0 95 291.6 305.0 304.6 255.3 256.5 265.0 247.7 254.2 251.4 100 286.8 301.6 302.7 246.6 257.3 251.2 245.2 245.9 247.2 CE 85 0.81 0.89 0.78 0.91 0.91 0.90 0.93 0.92 0.93 90 0.90 0.89 0.89 0.92 0.91 0.91 0.93 0.93 0.93 95 0.90 0.89 0.89 0.92 0.91 0.91 0.93 0.92 0.93 100 0.90 0.89 0.89 0.93 0.92 0.93 0.92 0.92 0.93 PI 85 0.61 0.92 0.90 0.81 0.93 0.96 0.85 0.94 0.97 90 0.79 0.92 0.95 0.83 0.94 0.96 0.85 0.95 0.97 95 0.80 0.92 0.95 0.84 0.94 0.96 0.85 0.95 0.97 100 0.80 0.92 0.95 0.85 0.94 0.97 0.84 0.94 0.97

Note: * “1, 2, and 3” denote one-, two-, and three-step-ahead forecasts; 1 “V” stands for the percentage of 1098 total variance. 1099

1100


73

Table 16. Multiple-step predictions for Wuxi and India series using LR, K-NN, and 1101

ANN with PCA2 1102

Watershed Performance

index V (%)1

LR K-NN ANN 1* 2* 3* 1 2 3 1 2 3

Wuxi RMSE 85 10.8 11.6 11.6 11.5 12.2 12.7 10.6 11.5 11.6 90 10.8 11.6 11.6 11.5 12.2 12.7 10.6 11.5 11.6 95 10.9 11.9 12.0 11.8 12.4 12.6 10.6 11.5 11.6 100 10.9 11.9 12.0 11.8 12.4 12.6 10.6 11.5 11.6 CE 85 0.14 0.01 0.00 0.02 -0.11 -0.19 0.16 0.02 0.00 90 0.14 0.01 0.00 0.02 -0.11 -0.19 0.16 0.02 0.00 95 0.12 -0.05 -0.07 -0.03 -0.14 -0.18 0.17 0.02 0.00 100 0.12 -0.05 -0.07 -0.03 -0.14 -0.18 0.16 0.02 0.00 PI 85 0.30 0.44 0.47 0.20 0.37 0.37 0.32 0.44 0.47 90 0.30 0.44 0.47 0.20 0.37 0.37 0.32 0.45 0.47 95 0.28 0.41 0.43 0.16 0.36 0.38 0.32 0.45 0.47 100 0.28 0.41 0.43 0.16 0.36 0.38 0.32 0.45 0.47 India RMSE 85 482.9 413.0 800.6 254.7 248.9 253.1 241.7 247.2 249.8 90 352.8 357.8 600.6 253.1 249.4 250.4 245.0 247.9 247.8 95 326.5 331.9 376.1 247.8 246.1 246.1 243.5 249.0 244.8 100 286.8 301.6 302.7 246.6 257.3 251.2 247.6 252.3 247.1 CE 85 0.73 0.80 -2.74 0.92 0.93 0.93 0.93 0.93 0.93 90 0.86 0.85 0.58 0.93 0.93 0.93 0.93 0.93 0.93 95 0.88 0.87 0.84 0.93 0.93 0.93 0.93 0.93 0.93 100 0.90 0.89 0.89 0.93 0.92 0.93 0.93 0.93 0.93 PI 85 0.44 0.86 -1.75 0.84 0.95 0.97 0.86 0.95 0.97 90 0.70 0.89 0.81 0.85 0.95 0.97 0.86 0.95 0.97 95 0.74 0.91 0.93 0.85 0.95 0.97 0.86 0.95 0.97 100 0.80 0.92 0.95 0.85 0.94 0.97 0.85 0.95 0.97

1103

1104


74

Table 17. Optimal p RCs for model inputs at various forecasting horizons 1105

Watershed Model Prediction horizons

Supervised filter (SSA1) Unsupervised filter (SSA2)

Optimal p RCs RMSE Optimal p RCs RMSE

Wuxi LR 1 1* 2* 6.01 1 2 6.01 2 1 5 7.73 1 5 7.73 3 1 8.40 1 8.40 K-NN 1 1 8.02 2 3 7.17 2 1 8.41 2 4 8.03 3 1 9.99 2 9.69 ANN 1 1 2 3 4.43 1 2 3 4.43 2 1 5 5.82 1 5 5.57 3 1 6.42 1 6.25 Zhenwan LR 1 1 2 3 7.19 1 2 3 7.19 2 1 7 2 7.99 1 2 7 7.99 3 1 5 8.81 1 5 8.81 K-NN 1 1 2 3 4 5 9.84 3 6 7.64 2 1 9.72 3 6 8.95 3 1 10.33 2 5 10.24 ANN 1 1 2 3 4 5.55 5 6 7 5.02 2 1 7 2 5.84 3 7 5.51 3 1 5 6.58 3 7 5.56 India LR 1 2 1 3 4 185.95 1 2 3 4 185.95 2 1 2 237.85 1 2 237.85 3 3 4 2 7 1 299.14 1 2 6 287.16 K-NN 1 2 1 3 4 5 236.90 1 2 5 6 236.39 2 1 2 247.44 1 2 5 242.65 3 3 4 2 7 249.43 1 2 5 6 243.86 ANN 1 2 166.58 1 7 164.70 2 1 2 7 173.57 1 2 7 166.30 3 3 4 2 7 1 235.59 1 5 7 172.56 Zhongxian LR 1 1 2 40.06 1 2 40.06 2 1 2 6 41.87 1 6 39.44 3 3 6 2 4 1 58.29 1 5 41.53 K-NN 1 1 2 51.78 1 2 51.78 2 1 2 53.86 1 2 3 53.32 3 3 6 2 54.15 2 3 53.16 ANN 1 1 2 3 38.39 1 5 6 34.09 2 1 44.49 1 5 6 39.45 3 3 6 46.14 1 5 34.94

Note:* the numbers of “1, 2” stand for RC1 and RC2, and these numbers in the SSA1 column is in a 1106 descending order of CCFs shown in Figure 6.12. 1107

1108


75

Table 18. Model performances at three forecasting horizons using MANN and ANN with the 1109

SSA2 1110

Watershed Model RMSE CE PI

1 2 3 1 2 3 1 2 3

WuXi ANN 4.43 5.57 6.25 0.84 0.78 0.70 0.87 0.88 0.84 MANN 3.63 4.32 3.93 0.90 0.86 0.89 0.92 0.92 0.94 Zhenwan ANN 5.02 5.51 5.56 0.81 0.75 0.71 0.88 0.85 0.84 MANN 3.18 3.20 3.31 0.92 0.92 0.91 0.95 0.95 0.95 India ANN 164.7 166.3 172.6 0.97 0.97 0.97 0.95 0.98 0.99 MANN 144.2 145.1 157.4 0.98 0.98 0.97 0.95 0.98 0.99 Zhongxian ANN 34.09 39.45 34.94 0.84 0.71 0.83 0.84 0.80 0.92 MANN 28.58 32.24 32.69 0.86 0.82 0.82 0.86 0.88 0.91

1111

1112


76

1113

Table 19. RMSE of the LR model coupled with SSA2 using various L 1114

Watershed Prediction horizons

L in SSA 3 4 5 6 7 8 9 10

Wuxi 1 6.13 5.94 6.01 6.41 5.83 5.81 5.61 5.51 2 7.79 7.62 7.73 8.14 7.71 7.75 7.62 7.66 3 11.76 8.61 8.40 9.04 9.23 8.72 8.56 8.62 Zhenwan 1 7.74 7.49 7.31 7.29 7.19 6.99 7.08 6.84 2 10.27 8.42 9.04 8.19 7.99 7.67 7.43 7.61 3 11.28 10.66 10.06 9.33 8.81 8.91 9.42 9.28

1115

1116

prediction of rainfall time series using modular

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